Abstract
In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus [Formula: see text]. We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321] and Rado [On Plateau’s problem, Ann. Math. 31 (1930) 457–469]. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-[Formula: see text] minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding–Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory by Colding–Minicozzi and the author to all genera.
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